Unformatted text preview: 2 MA 36600 MIDTERM #1 REVIEW §1.3: Classiﬁcation of Diﬀerential Equations.
• There are two types of diﬀerential equations:
– ODEs or ordinary diﬀerential equations are equations which do not involve partial derivatives.
– PDEs or partial diﬀerential equations are equations which do involve partial derivatives.
• Consider a function y = y (t), and let y (k) denote the k th derivative. An nth order diﬀerential
equation is an ordinary diﬀerential equation in the form
F t, y, y (1) , y (2) , . . . , y (n) = 0
y (n) = G t, y, y (1) , y (2) , . . . , y (n−1) .
• We say that this diﬀerential equation is linear if G(t, ) = g0 (t) + g1 (t) y1 + g2 (t) y2 + · · · + gn (t) yn
y is a linear function for the vector = (y1 , y2 , . . . , yn ). Otherwise, we say that this diﬀerential
equation is nonlinear.
§2.1: Linear Equations; Method of Integrating Factors.
• An ordinary diﬀerential equation is said to be a ﬁrst order equation if it is a relation in the form
F (t, y, y ) = 0 =⇒ dy
= G(t, y ).
dt We say that this ﬁrst order equation is linear if
G(t, y ) = g (t) − p(t) y =⇒ dy
+ p(t) y = g (t).
dt • Say that we can ﬁnd a function µ = µ(t) such that
+ µ(t) p(t) y =
µ(t) y .
Such a function is called an integrating factor. More precisely, it is a function such that
i. µ(t) is not identically zero, and
ii. µ (t) = p(t) µ(t).
• The general solution of y + p(t) y = g (t) is the function
µ(τ ) g (τ ) dτ + C
µ(t) = exp
p(τ ) dτ .
y (t) =
µ(t) §2.2: Separable Equations.
• Consider an ordinary diﬀerential equation in the form y = G(t, y ). We say that it is separable if we
have a factorization
G(t, y ) = Y (y ) T (t).
• To solve such an equation, the trick is to bring all of the terms involving y to one side, and all of
the terms involving t to the other:
= Y (y ) T (t)
= T (t)
dy = T (t) dt + C.
Y (y ) dt
Y (y )
The general solution is f (t, y ) = C
where f (t, y ) = f1 (y ) − f2 (t) in terms of the antiderivatives
f1 (y ) =
f2 (t) =
T (τ ) dτ .
Y (σ ) • The expression “f (t, y ) = C ” is an implicit solution to the diﬀerential equation y = G(t, y ). The
constant C can be found from the initial conditions. The graphs of the expressions f (t, y ) = C in
the ty -plane are called integral curves. ...
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